Testing for unit roots and non-linear transformations

It is well known that the augmented Dickey–Fuller (ADF) test of unit roots in univariate time series is sensitive to non-linear transformations: a common example is when variables expressed in logarithms are found to be stationary, whereas the same variables in levels are found to be non-stationary. In this paper, the effects of non-linear transformations on the ADF auxiliary regression are investigated within the class of the Box–Cox model, and a test of non-linear transformation is developed to assess the adequacy of the ADF regression. The proposed test is computationally simple and is calculated as thet ratio of an added variable in the ADF regression. Cointegration among series which are subject to non-linear transformations is also analysed, and a simple procedure is developed to test the non-linear transformation used in cointegration analysis. Several empirical examples taken from the Nelson–Plosser data set illustrate the practical relevance of the proposed test for univariate series, and a second empirical example is used to illustrate the test of non-linear transformation for cointegrated series

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